Question

how to remember all the trigonometric functions of sum and difference of two angles without rote memorizing it!

Answer 1

The best way is to practice as many questions as you can. The more you do, the more you remember. Don't always look at the formulas from the book. As you solve, try to remember and it will get easier.
As you solve, see the patterns and try to visualise them.

For example:
sin(A+B) = sinAcosB+cosBsinA
cos(A+B) = cosAcosB+sinAsinB
Here, you see that for both, formula contains summation of two terms and each term is multiplication of two terms. It is common in both. What is different?
for sin - sin and cos terms are mixed(sinAcosB and cosAsinB)
for cos - sin and cos terms are separate (cosAcosB and sinAsinB)

You form your own patterns which you can remember.